Designing Wormholes in Novel Power-Law $f(R)$: A Mathematical approach with a linear equation of state

TL;DR

This work develops four independent $f(R)$ wormhole models guided by a novel linear equation of state in the Morris–Thorne geometry, identifying both NEC-satisfying and NEC-violating regimes. By solving a master equation that couples $f(R)$, shape, and redshift functions, the authors obtain explicit power-law $f(R)$ forms and analyze their geometric, energy-condition, and scalar-tensor representations. They demonstrate the existence of wormholes with ghost or non-ghost $f(R)$ sectors, establish equivalence to Brans–Dicke theory with $\omega=0$, and validate robustness via hybrid metric–Palatini gravity, while also locating photon spheres and linking them to the Herrera complexity factor. The results show that the essential relation between complexity and photon spheres persists in $f(R)$ gravity, and they discuss cosmological viability for the non-ghost models, highlighting potential applications in cosmology and astrophysical lensing.

Abstract

We consider the inhomogeneous Morris-Thorne wormhole metric with matter tensors characterised by a novel linear equation of state in $f(R)$ gravity. Using the Einstein's field equations in metric $f(R)$ gravity we model solutions for both wormhole as well as $f(R)$ gravity. We obtain four different wormhole models, two wormholes are characterised by solid angle deficit, three are not asymptotically extendible, while one is asymptotically flat with zero tidal force. These are supported by four different power law $f(R)$ models. The parameter space of the models can support both null energy conditions (NEC) satisfying as well as violating wormhole. In case of NEC satisfying matter, the associated $f(R)$ is ghost. The $f(R)$ models obtained have been independently substantiated for cosmological feasibility and valid parameter space was obtained corresponding to cosmologically viable $f(R)$. Suitable scalar-tensor representation of the corresponding $f(R)$ models have been presented using the correspondence of $f(R)$ gravity with Brans-Dicke (BD) theory of gravity. The robustness of the wormhole solutions were further analysed with the BD scalar fields in the hybrid metric-Palatini gravity, which showed excellent results. Lastly as an independent astrophysical probe for the wormhole we have obtained the location of their photon spheres and have connected them with the Herrera Complexity factor in $f(R).$ Our results show that the relation between the complexity factor and existence of photon spheres remains fundamentally unaltered in $f(R)$ as compared to Einstein's gravity.

Figures (6)

• Figure 1: Figures (a) and (b) respectively depict the embedding diagram for the wormhole throat and the $f(R)$ as a function of the curvature $R$ corresponding to Solution 1. The figures have been obtained for $r_{0}=1,~\delta=-6,~\alpha=-1$ and $f_{1}=1.$ Figures (c) and (d) respectively show the embedding for the wormhole throat and the $f(R)$ solution as a function of $R$ corresponding to Solution 2. Both (c) and (d) have been constructed with parameters $r_{0}=1,~\delta=3,~f_{2}=1.$
• Figure 2: Figures (a) and (b) respectively depict the embedding diagram for the wormhole throat and the $f(R)$ as a function of the curvature $R$ corresponding to Solution 3. The figures have been obtained for $r_{0}=1,~\delta=-4,~\alpha=-1$ and $f_{1}=1.$ Figures (c) and (d) respectively show the embedding for the wormhole throat and the $f(R)$ solution as a function of $R$ corresponding to Solution 4. Both (c) and (d) have been constructed with parameters $r_{0}=1,~\delta=3,~f_{4}=-1.$
• Figure 3: (a) The energy conditions corresponding to Solution 1, figure drawn corresponding to $r_{0}=1,~\delta=-8,~\alpha=1,~f_{1}=1.$ (b) The energy conditions corresponding to Solution 2, figure drawn corresponding to $r_{0}=1,~\delta=4,~f_{2}=1.$ (c) The energy conditions corresponding to Solution 3, figure drawn corresponding to $r_{0}=1,~\delta=-6,~\alpha=1,~f_{1}=1.$ (d) The energy conditions corresponding to Solution 4, figure drawn corresponding to $r_{0}=1,~\delta=3,~f_{4}=-1.$
• Figure 4: (a) Evolution of $f(R)$ wrt $R$ of Solution 1 in the parameter validity regions I-III. (b) Evolution of $f(R)$ wrt $R$ of Solution 3 in the parameter validity regions IV and V. Both figures were drawn for $r_{0}=1,~f_{1}=1.$
• Figure 5: Figures (a)-(d) give the evolution of the Potential function $W(\phi)$ wrt the BD scalar $\phi$ corresponding to the Solutions 1-4 respectively. The parameters were chosen such that they satisfy the cosmological viability of the $f(R)$ models. Table 1 provides the parameter inclusions for cosmological viability of $f(R)$ models 1-4.